For the sake of convenience, the current description focuses on systems and techniques that may be realized in a particular embodiment of cantilever-based instruments, the atomic force microscope (AFM). Cantilever-based instruments include such instruments as AFMs, scanning probe microscopes, molecular force probe instruments (1D or 3D), high-resolution profilometers (including mechanical stylus profilometers), surface modification instruments, chemical or biological sensing probes, and micro-actuated devices. The systems and techniques described herein may be realized in such other cantilever-based instruments as well as AFMs.
An AFM as shown in FIG. 1 is a device used to produce images of the topography of the sample (and/or other sample characteristics) based on information obtained from scanning (e.g., rastering) a sharp tip on the end of a cantilever 1010 attached to a chip 1030 relative to the sample surface. Topographical and/or other features of the sample are detected by detecting changes in deflection and/or oscillation 1040 characteristics of the cantilever (e.g., by detecting small changes in deflection, phase, frequency, etc., and using feedback 1060 to return the system to the reference state). By scanning the probe relative to the sample surface 1090, a “map” of the sample topography or other sample characteristics may be obtained.
Changes in deflection or oscillation of the cantilever are typically detected by an optical lever arrangement whereby a light beam is directed onto the back of the cantilever opposite the tip 1010. The beam reflected from the cantilever illuminates a spot on a position sensitive detector (PSD 1020). As the deflection or oscillation of the cantilever changes, the position of the reflected spot on the PSD changes, causing a change in the output from the PSD. In addition changes in the deflection or oscillation of the cantilever are typically made to trigger a change in the vertical position of the cantilever base relative to the surface of the sample (referred to herein as a change in the Z position, where Z is generally orthogonal to the XY plane defined by the sample surface), in order to maintain the deflection or oscillation at a constant pre-set value 1050. This feedback 1060 is typically used to generate an AFM image 1110.
Actuators 1080 are commonly used in AFMs, for example to raster the cantilever or to change the position of the cantilever base relative to the surface of the sample. The purpose of actuators is to provide relative movement between different components of the AFM; for example, between the probe 1010 and the sample 1040. For different purposes and different results, it may be useful to actuate the sample, the cantilever or the tip of the cantilever, or some combination of these elements. Sensors are also commonly used in AFMs. They are used to detect movement, position, or other attributes of various components of the AFM, including movement created by actuators.
For the purposes of the specification, unless otherwise indicated, the term “actuator” 1080 refers to a broad array of devices that convert input signals into physical motion, including piezo activated flexures, piezo tubes, piezo stacks, blocks, bimorphs, unimorphs, linear motors, electrostrictive actuators, electrostatic motors, capacitive motors, voice coil actuators and magnetostrictive actuators, and the term “sensor” or “position sensor” refers to a device that converts a physical parameter such as displacement, velocity or acceleration into one or more signals such as an electrical signal, including capacitive sensors, inductive sensors (including eddy current sensors), differential transformers (such as described in co-pending applications US20020175677A1 and US20040075428A1, Linear Variable Differential Transformers for High Precision Position Measurements, and US20040056653A1, Linear Variable Differential Transformer with Digital Electronics, which are hereby incorporated by reference in their entirety), variable reluctance, optical interferometry, optical deflection detectors (including those referred to above as a PSD and those described in co-pending applications US20030209060A1 and US20040079142A1, Apparatus and Method for Isolating and Measuring Movement in Metrology Apparatus, which are hereby incorporated by reference in their entirety), strain gages, piezo sensors, magnetostrictive and electrostrictive sensors.
AFMs can be operated in a number of different sample characterization modes, including contact mode where the tip of the cantilever is in constant contact with the surface of the sample, and AC modes where the tip makes no contact or only intermittent contact with the sample surface.
In both the contact and ac 1080 sample characterization modes, the interaction between the tip of the cantilever and the sample induces a discernable effect on a cantilever-based operational parameter, such as the cantilever deflection, the cantilever oscillation amplitude or the frequency of the cantilever oscillation, the phase of the cantilever oscillation relative the signal driving the oscillation, all of which are detectable by a sensor. AFMs use the resultant sensor-generated signal as a feedback control signal for the Z actuator to maintain constant a designated cantilever-based operational parameter.
To get the best resolution measurements, one wants the tip of the cantilever to exert only a low force on the sample. In biology, for example, one often deals with samples that are so soft that forces above 10 pN can modify or damage the sample. This also holds true for high resolution measurements on hard samples such as inorganic crystals, since higher forces have the effect of pushing the tip into the sample, increasing the interaction area and thus lowering the resolution. For a given deflection of the probe, the force increases with the spring constant (k) of the cantilever. When operating in air in AC modes where the tip makes only intermittent contact with the sample surface, spring constants below 30 N/m are desirable. For general operation in fluid, very small spring constants (less then about 1.0 N/m) are desirable.
At the same time it is often useful in biology to measure the stiffness of a sample, to distinguish DNA from a salt crystal for example. Images of the topography of a sample do not tell us much about stiffness.
With contact AFM it has been common to measure the interaction forces between the cantilever and the sample with Hooke's Law, a relationship describing the behavior of springs, where the force exerted by the spring, F, is equal to a constant characterizing the spring, k, times a change in position of the spring, x: F=kx. In the case of AFMs, the spring is the cantilever, the constant is the spring constant of the cantilever, and the change in position is a change in the deflection of the cantilever as measured by the PSD.
Early spring constant calculations were based on order-of-magnitude knowledge. One of the first attempts to produce a more accurate determination has come to be known as the Cleveland method, after Jason Cleveland, then a graduate student at the University of California, Santa Barbara. Cleveland, J. P., et al., A nondestructive method for determining the spring constant of cantilevers for scanning force microscopy, Rev. Scientific Instruments 64, 403, 1993. The Cleveland method estimates the spring constant by measuring the change in resonant frequency of the cantilever after attaching tungsten spheres of known mass to the end of the cantilever. Cleveland claims that this method should be applicable to most cantilevers used in calculating force.
A few years after publication of the Cleveland method, a simpler method for estimating the spring constant noninvasively was published which has also come to be known under the name of the lead author, John Sader. Sader, John E., et al., Calibration of rectangular atomic force microscope cantilevers, Rev. Scientific Instruments 70, 3967, 1999. The theory starts with the well-known relationship between stiffness, mass, and resonance frequency (k=mω2) which provides an intuitive way to calibrate the stiffness of a cantilever in air by simply measuring its width, length, height and resonance frequency. However, the large error in the thickness of the cantilever, which may also not be uniform, leads to a poor estimate of the cantilever mass. Whereas modeling the inertial loading of the cantilever (the cantilever mass) is inaccurate, the viscous loading of the cantilever can be reliably modeled in fluids such as air. In ambient conditions, the viscous loading is dominated by hydrodynamic drag of the surrounding air. This hydrodynamic drag depends on the density and viscosity of air, and the plan-view geometry of the cantilever. Because the thickness of the cantilever plays no role in the hydrodynamic drag, the viscous loading can be very accurately modeled for any given cantilever shape and used to calibrate cantilevers. In summary, the Sader method provides a straightforward formula for estimating the spring constant of a rectangular cantilever which requires the resonant frequency and quality factor of the fundamental mode of the cantilever, as well as its plan view dimensions. The method also assumes that the quality factor is equal to or greater than 1, which is typically true if the measurement is made in air.
The spring constant of the cantilever provides the information necessary to calculate the force exerted by the cantilever. In an AFM system where the sample is moved relative to the cantilever, the additionally required information is the change in the deflection signal of the cantilever (as measured by the PSD in volts), as a function of the distance the actuator moves the sample (which is a conversion into distance units of the voltage applied to the actuator). The estimate of this relationship is often referred to as the calibrated sensitivity of the optical lever of the AFM, or just optical lever sensitivity (“OLS”).
Where a relatively rigid sample was available, early estimates of the sensitivity of the optical lever of an AFM were typically made by pushing the tip of the cantilever into the sample and using the voltage response of the PSD (which is a measure of the change in the deflection of the cantilever, but since it is in volts not a measure that is directly usable for our purposes) taken together with the distance the actuator moves the sample to estimate the factor that can be applied to convert other voltage measurements of deflection into distance. This method has some disadvantages: (1) where the sample is soft, as is typically the case with biological samples, the response to pushing the tip into the sample will be nonlinear and therefore not useful for estimating the voltage response of the PSD or the distance the actuator moves the sample; (2) where the sample is a biological sample, it may contaminate the tip when the tip is pushed into the sample and here too yield a nonlinear response; and (3) it is altogether too easy to damage the tip of a cantilever when pushing it into a sample.
An xy graph can be used to show typical sample displacement on the x-axis (the more the sample is raised the further out we are on the x-axis) and tip displacement on the y-axis (the more the tip is raised the further out on the y-axis) when the sample is raised toward the tip during the process of making an estimate of the sensitivity of the optical lever of an AFM. At point A on the graph the sample and the tip are sufficiently far apart that neither exerts a force on the other. When the sample is raised from point A to point B however the distance from the tip is sufficiently small that the attractive Van der Waals force is operative and the tip is pulled toward the surface. At point C the Van der Waals force has overcome the spring tension of the cantilever and the tip has dropped to the surface of the sample. If we continue to raise the sample further at this point, the tip will remain in contact with the rising sample following it upward along the path designated by points C, E and F. If the sample is retracted from point F the tip will again follow—even beyond point C where the tip had dropped to the surface of the sample when the sample was being raised (the extent the tip follows the sample will depend in part on the presence of capillary forces, a normal condition except when the AFM is operating in a vacuum or in liquids). At some point however the tip will break free from the sample and return to the null position line. The point at which the tip breaks free is point G.
The distance on the y-axis from point G to point F is an estimate of the change in the deflection of the cantilever when the tip of the cantilever is pushed into the sample, just as the distance on the x-axis between those points is an estimate of the distance the actuator moves the sample. The relation between these estimates gives us the factor that can be applied to convert other voltage measurements of deflection into distance. As noted above this factor is usually referred to as optical lever sensitivity, or OLS. In order to distinguish it from a similar factor to be discussed below we will also call it the dc OLS.
Recently a group working a Trinity College, Dublin and Asylum Research in California, as well as John Sader, have developed a method which provides a straightforward formula for estimating OLS noninvasively. M. J. Higgins, et al., Noninvasive determination of optical lever sensitivity in atomic force microscopy, Rev. Scientific Instruments 77, 013701, 2006. The method requires the thermal noise spectrum of the cantilever to be measured (on an AFM) and the fundamental mode of the spectrum to be fitted to the power response function of the simple harmonic oscillator. The equipartition theorem is then applied to this result and after simplification the result is a straightforward formula for estimating inverse dc OLS which requires the spring constant of the cantilever, as well as its resonant frequency, quality factor and dc power response.
We have discussed above the dc OLS factor estimated by raising and retracting the sample relative to the tip of the cantilever and comparing the deflection of the cantilever as measured on the y-axis with the distance the actuator moves the sample as measured on the x-axis. The dc OLS factor pertains to operation of an AFM in the contact mode. In ac modes a different OLS factor may be estimated. In ac modes the cantilever is initially oscillating at its free amplitude unaffected by the sample. In an AFM system where the sample is moved relative to the cantilever, as the sample is raised to the oscillation path however the tip of the oscillating cantilever contacts the sample and the amplitude of the vibration decreases. Initially for each nanometer the sample is raised the amplitude of the vibration decreases by a nanometer as well. When the sample begins to retract the amplitude increases until the tip is free of the sample surface and the amplitude of the cantilever oscillation levels off at its free amplitude. The ac OLS factor is estimated by comparing the cantilever deflection amplitude as measured on the y-axis for one complete extension and retraction of the sample actuator with the distance the actuator moves the sample as measured on the x-axis. As with the dc OLS factor, cantilever deflection amplitude is measured by the PSD in volts and the distance the actuator moves the sample is a conversion into distance units of the voltage applied to the actuator.
As with the spring constant and dc OLS, a noninvasive method for measuring the ac OLS factor has also been developed. R. Proksch, et al., Finite optical spot size and position corrections in thermal spring constant calibration, Nanotechnology 18, 1344, 2004. The authors make the point, as had others before them, that the value of OLS will be different depending on whether the cantilever is “deflected by a localized and static force at the end [that is the end where the tip is located], as in the case of force measurements or AFM imaging, or whether it vibrates freely . . . .” They derive a factor for relating the two OLSs given by κ=InvOLSfree/InvOLSend. It will be noted that the authors' invOLSfree measurement is essentially the ac OLS measurement discussed above and the InvOLSend measurement is essentially the dc OLS measurement discussed above. In their discussion of the derivation of the factor for relating the two OLSs the authors make the point that the factor is dependent on the location of the laser spot on the cantilever.